Remark 3.4 
We would like to emphasize that Proposition REF 
is not a consequence of the general theory of 2-cocycle deformations
from [1]}, [2]} and [3]}.
In such a setting, it is convenient to observe that
the \(K\) -coaction on \(A\)  induces to associated modules,
i.e., \(E_A \colon {}^H\!\!{M} \rightarrow {}_{B}^{K}{M}\) 
can be regarded as a functor to the category
of left \(K\) -comodule left \(B\) -modules.
Consequently, the deformed associated module functor
\(E_{A_\theta } \colon {}^H\!\!{M} \rightarrow {}_{B_\theta }^{K_\theta }{M} \) 
can be regarded as a functor to the category
of left \(K_\theta \) -comodule left \(B_\theta \) -modules,
where \(K_\theta \)  and \(B_\theta \)  are the 2-cocycle deformations of
\(K\)  and \(B\) . (In our specific example of interest, we have that \(K_\theta =K\) 
and \(B_\theta =B\)  (cf. Lemma REF ),
but we shall keep these labels to make the discussion below more transparent.)
Using the 2-cocycle deformation functor
from [2]} or [3]},
which we shall denote by \(\Sigma _\theta \) , one obtains a commutative diagram
\(@C=4em{[rd]_-{E_{A_\theta }} {}^H\!\!{M} [r]^-{E_A} & {}_{B}^{K}{M}[d]^-{\Sigma _\theta }\\& {}_{B_\theta }^{K_\theta }{M}}\) 
